On near-rings with two-sided $alpha$-derivations

Abstract

In this paper, we introduce the notion of two-sided a-derivation of a near-ring and give some generalizations of [1]. Let N be a near ring. An additive mapping f: $Nrightarrow N$ is called an $(alpha,beta)$ -derivation if there exist functions $alpha,beta :Nrightarrow N$ such that f(xy)=f(x)$alpha$(y)+$beta$ (x)f(y) for all x,y $ in$ N. An additive mapping d:$Nrightarrow N$ is called a two-sided $alpha$ -derivation if d is an $(alpha,1)$-derivation as well as a (1,$ alpha$)-derivation. The purpose of this paper is to prove the following two assertions: (i) Let N be a semiprime near-ring, I be a subset of N such that $0in I$, IN$subseteq$I and d be a two-sided $alpha$ -derivation of N. If d acts as a homomorphism on I or as an anti-homomorphism on I under certain conditions on a, then d(I)= {0}. (ii) Let N be a prime near-ring, I be a nonzero semigroup ideal of N, and d be a $(alpha, 1)$ -derivation on N. If d+d is additive on I, then (N,+) is abelian.